How to use Image Mapping to repeat cut-in-half textures with X and Y in Cycles?
To mirror the texture you can split the UV Coordinates using a Separate XYZ node and then manipulate them using Maths nodes.

The UV and Image Texture range is from 0.0 to 1.0 and so you can mirror one side by Subtracting 0.5 (so it's from -0.5 to 0.5), applying Absolute (so negatives become positive) and Adding the 0.5 back in (so the range ends at 1.0) as follows :



To mirror the opposite side you can simply add another Subtract node to subtract the result from 1.0 to reverse the range :





EDIT The above solution assumes that the UV map is covering the entire 'face' of your model - ie, with the 'mirror' at 0.5 of the UV map. If this is not the case (eg, if the centre line is actually at 1.0) then you can simply use Mapping nodes to shift/scale the range before the mirroring and shift/scale back again afterwards.

For example, consider the following UV map for Suzanne :



Note how the 'centre-line' is at U = 1.0, rather than 0.5. For this we can use the following nodes to achieve the mirroring :



Note how the first mapping nodes scales by 0.5 in the X direction (note that it's in 'Point' mode - if using 'Texture' node, invert the scale to 1/0.5 = 2.0 instead). The final mapping node reverses the scale.

This produces the following result :